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  1.  15
    Consequences of Arithmetic for Set Theory.Lorenz Halbeisen & Saharon Shelah - 1994 - Journal of Symbolic Logic 59 (1):30-40.
    In this paper, we consider certain cardinals in ZF (set theory without AC, the axiom of choice). In ZFC (set theory with AC), given any cardinals C and D, either C ≤ D or D ≤ C. However, in ZF this is no longer so. For a given infinite set A consider $\operatorname{seq}^{1 - 1}(A)$ , the set of all sequences of A without repetition. We compare $|\operatorname{seq}^{1 - 1}(A)|$ , the cardinality of this set, to |P(A)|, the cardinality of (...)
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  2.  9
    On Shattering, Splitting and Reaping Partitions.Lorenz Halbeisen - 1998 - Mathematical Logic Quarterly 44 (1):123-134.
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  3.  17
    Relations Between Some Cardinals in the Absence of the Axiom of Choice.Lorenz Halbeisen & Saharon Shelah - 2001 - Bulletin of Symbolic Logic 7 (2):237-261.
    If we assume the axiom of choice, then every two cardinal numbers are comparable, In the absence of the axiom of choice, this is no longer so. For a few cardinalities related to an arbitrary infinite set, we will give all the possible relationships between them, where possible means that the relationship is consistent with the axioms of set theory. Further we investigate the relationships between some other cardinal numbers in specific permutation models and give some results provable without using (...)
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  4.  6
    Mathias Absoluteness and the Ramsey Property.Lorenz Halbeisen & Haim Judah - 1996 - Journal of Symbolic Logic 61 (1):177-194.
    In this article we give a forcing characterization for the Ramsey property of Σ 1 2 -sets of reals. This research was motivated by the well-known forcing characterizations for Lebesgue measurability and the Baire property of Σ 1 2 -sets of reals. Further we will show the relationship between higher degrees of forcing absoluteness and the Ramsey property of projective sets of reals.
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  5.  17
    Symmetries Between Two Ramsey Properties.Lorenz Halbeisen - 1998 - Archive for Mathematical Logic 37 (4):241-260.
    In this article we compare the well-known Ramsey property with a dual form of it, the so called dual-Ramsey property (which was suggested first by Carlson and Simpson). Even if the two properties are different, it can be shown that all classical results known for the Ramsey property also hold for the dual-Ramsey property. We will also show that the dual-Ramsey property is closed under a generalized Suslin operation (the similar result for the Ramsey property was proved by Matet). Further (...)
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  6.  8
    A Generalization of the Dual Ellentuck Theorem.Lorenz Halbeisen & Pierre Matet - 2003 - Archive for Mathematical Logic 42 (2):103-128.
    We prove versions of the Dual Ramsey Theorem and the Dual Ellentuck Theorem for families of partitions which are defined in terms of games.
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  7.  1
    A Weird Relation Between Two Cardinals.Lorenz Halbeisen - forthcoming - Archive for Mathematical Logic:1-7.
    For a set M, let \\) denote the set of all finite sequences which can be formed with elements of M, and let \ denote the set of all 2-element subsets of M. Furthermore, for a set A, let Open image in new window denote the cardinality of A. It will be shown that the following statement is consistent with Zermelo–Fraenkel Set Theory \: There exists a set M such that Open image in new window and no function Open image (...)
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  8.  1
    Making Doughnuts of Cohen Reals.Lorenz Halbeisen - 2003 - Mathematical Logic Quarterly 49 (2):173-178.
    For a ⊆ b ⊆ ω with b\ a infinite, the set D = {x ∈ [ω]ω : a ⊆ x ⊆ b} is called a doughnut. A set S ⊆ [ω]ω has the doughnut property [MATHEMATICAL SCRIPT CAPITAL D] if it contains or is disjoint from a doughnut. It is known that not every set S ⊆ [ω]ω has the doughnut property, but S has the doughnut property if it has the Baire property ℬ or the Ramsey property ℛ. (...)
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